Begin with the chorus, unless you already have ideas for the rest of the composition. The chorus is generally the catchiest, most memorable section, so it often makes sense to write it first. To make the chorus catchy, try keeping the phrases short and including a noticeable amount of repetition. Also, the chorus usually has some of the highest notes, if not the highest note. (Notice how the pitch, along with the emotional intensity, jumps to a higher level in the chorus of "Hey There Delilah.") Another trick is to write the chorus based on a cell. A cell is a small group of notes---three, four or perhaps five---that either are used exclusively or at least primarily. For example, if you are writing in the key of C major, you might select the notes C, D, F and A. Ironically, restricting your choices in this manner tends to stimulate creativity.
Write the melody to be used for the verses. Keep in mind that the verses should contrast with the chorus and yet be compatible with it. One way you can do this is to select a different cell to use, yet one which has notes in common with the cell of the chorus. The rhythm also should contrast; it's particularly appropriate to have the verse begin with shorter notes and the chorus begin with longer notes, providing more emphasis. (Notice this technique in "Hey There Delilah.") It is often effective, particularly in instrumental music, to add a pre-chorus or "climb" to make a transition from the verse to the chorus. This is a very short phrase, perhaps only a couple of measures, and it often literally climbs in pitch to reach the emotional plateau of the chorus.
Write the bridge, which offers even stronger contrast---you might even want to modulate to a different key. In fact, in some cases, the bridge simply repeats the same melody as the verse or chorus in a different key (and usually at a higher pitch). Eight measures is the typical length for a bridge, and therefore, it is often referred to as a "middle eight." Notice that in "Hey There Delilah," it is the bridge, rather than the chorus, that contains the highest notes in the song. This is very effective, creating tension to be resolved in the verse that follows.
Tinker with the exact sequence if you want once you have the building blocks established. Rather than state the verse twice the first two times, and then only once, you might want to state it twice the first time, and then only once the second two times. Rather than repeat both the verse and chorus after the bridge, you might repeat only the chorus. And for a long piece, you might even consider two bridges.